PREDICTION OF AIRFOIL BEHAVIOR

In Chapter Two it was noted that the concepts of a point vortex and a point source could be extended to a continuous distribution of the elementary flow functions. In that chapter a distribution of sources in a uniform flow was found to produce a nonlifting body of finite thickness. In the case of the circular cylinder, the addition of a vortex also produced lift.

Comparable to the continuous distribution of sources pictured in Figure 2.20, consider a similar distribution of vortices as illustrated in Figure 3.13. Such a distribution is referred to as a vortex sheet. If у is the strength per unit length of the sheet, у Ax will be the total strength enclosed by the dashed contour shown in the figure. The contour is taken to lie just above and below the sheet. Ax is sufficiently small so that the velocity tangent to the sheet, v, can be assumed to be constant. Because of the symmetry to the flow provided by any one segment of the sheet, the tangential velocity just below the sheet is equal in magnitude but opposite in direction to that just above the sheet. From Equation 2.55, relating circulation to the strength of a vortex, it follows that

у Ax = 2v Ax or

Note the similarity of this relationship to that expressed by Equation 2.82. However, in the case of Equation 3.14, the velocity is tangent to the vortex sheet whereas, for Equation 2.82, the velocity is normal to the line on which the sources lie.

Consider now the thin airfoil pictured in Figure 3.14. If the airfoil is producing a lift, the pressure on the lower surface is greater than that on the upper. Hence, from Bernoulli’s equation, the velocity on the upper surface is greater than the velocity on

).

Figure 3.14 The velocity difference across a lifting thin airfoil.

velocity across the airfoil equal 2v, the upper and lower velocities can be written as

Vu = V+v and

V, = V-v

Thus the flow field around the airfoil is the same as that which would be produced by placing, in a uniform flow of velocity V, a vortex sheet of unit strength 2v along the airfoil.

The contribution to the lift of a differential length of the airfoil will be

dl=(pi – pu) dx

Or, using Bernoulli’s equation, this becomes,

dl = pV(2v) dx

Since 2v is the unit vortex strength, the Kutta-Joukowski law (Equation 2.81) is found to hold for the airfoil element.

dl = pVy dx

or, integrating the above equation over the entire chord,

l = pVT (3.15)

where Г is the total circulation around the airfoil given by

In order to predict the lift and moment on the airfoil, one must find the chordwise distribution of y(x) that will produce a resultant flow everywhere tangent to the mean camber line (thin airfoil approximation). In addition, the Kutta condition is applied at the trailing edge to assure that the flow leaves the trailing edge tangent to the mean camber line at that point. This is a

necessary condition; otherwise, the resulting flow will appear similar to Figure 3.4a with the lift being equal to zero.

An analytical solution to the thin airfoil will be obtained later but, first, let us consider a numerical approach to predicting the lift and moment of an airfoil.

As a gross approximation to the distributed vorticity along the airfoil, the distribution will be replaced by only one vortex of unknown strength, Г. However, Г will be placed at a particular point on the airfoil, at the quarter- chord point. The boundary condition and the Kutta condition will be satisfied at only one point, the three-quarter-chord point. This approximation, known as Weissinger’s approximation, is illustrated in Figure 3.15 for a flat-plate airfoil.

The velocity induced at 3c/4 by Г placed at cl4 will be

Г

Vj = —

TTC

Assuming a to be a small angle, it follows that

Vi = Via or

Г = ire Vat

m■

From the Kutta-Joukowski relationship, L = рУГ, so that

L = prrcV2a (3.19)

Expressing lift in terms of the lift coefficient and using Equation 3.19 leads to

Ci = lira (3.20)

where a is the angle of attack in radians.

The expression agrees identically with the theoretical solution of this problem that follows. Notice that the result predicts the slope of the lift curve, dCJda, to be 2ir/rad. Experimentally this figure is usually found to be

Flgun 3.15 Weissinger’s approximation to a thin airfoil.

somewhat less. Figures 3.5 and 3.6, for example, show a value of around

0. 105/deg or 6.02/rad.

The approximation of Figure 3.15 can be improved on by dividing the airfoil chord into a number of equal segments and placing a vortex of unknown strength at the quarter-chord point of each segment. The unknown strengths are determined by assuring that the normal velocity vanishes at the three-quarter-chord point of each segment. With the last control point down­stream of the last vortex singularity, the Kutta condition is assured.

To illustrate this numerical solution of the thin airfoil, consider Figure

3.16. Here, a circular arc airfoil having a unit chord length with a maximum camber ratio of z is operating at an angle of attack a.

If it is assumed that

z * 1 (3.21)

the radius of curvature, R, of the airfoil will be related to z approximately by

The slope of the camber line relative to the chord line (the angle ф in Figure 3.16) at any distance x can be determined from the geometry of the figure.

0 = (4-8x)z (3.22)

figure 3.16 A circular arc airfoil approximated by two vortices.

The component of V normal to the mean camber line and directed upward is thus

vn = V(a – ф)

It follows that at control points 1 and 2 located at x values of 3c/8 and 7cIS, respectively, the two vortices similating the airfoil must induce velocities downward given by

Vi = V(a – z) at 1 (3.23a)

Vi = V(a + 3z) at 2

The problem is linearized by Equation 3.21 so that the vortices, – yi and y2, are taken to lie on the chord line. Thus, according to Equation 2.56, the total velocities induced at the two control points by the two vortices will be

Applying the Kutta-Joukowski law to each vortex results not only in a predicted total lift, but also in a moment. In coefficient form the lift and moment (about the leading edge) become

(3.26a)

(3.26b)

The moment coefficient about the leading edge can be transferred to the quarter-chord point by using

С = c

C"»LE

Thus,

This simple, two-point model results in several important observations that are in agreement with more exact solutions. First, note that Equation 3.26a shows the lift coefficient to be a linear combination of a and z. Thus,
cambering an airfoil will not change the slope of the lift curve. Second, it is predicted that the moment about the quarter chord will be independent of a. Hence, this point is predicted to be the aerodynamic center.

As one divides the airfoil into a greater and greater number of elements, the resulting у distributions will approach the theoretical pressure distribution predicted on the basis of continuous у distributions. The strength, y, of a vortex placed at the c/4 point of an element of length Ax will be related to the pressure jump, Дp, across the element by

(3.28)

Figure 3.17 presents a comparison, for the flat-plate airfoil, between the pressure distribution obtained using the foregoing numerical procedure with that based on a continuous distribution of у along the chord. It is seen that the numerical results rapidly converge to the continuous solution as the number of elements increases. In preparing this figure it should be noted that Дp, given by Equation 3.28, has been expressed in coefficient form and plotted at the location of each point vortex. Figure 3.18 presents a similar comparison for the circular arc airfoil. In this case a is taken to be zero, avoiding the infinitely negative Cp at the leading edge.

Figure 3.17 Comparison of numerical calculation of chordwise lift distribution with analytical prediction for a flat-plate airfoil at 10“ angle of attack.

0.7 i—

The numerical model predicts the lift in exact agreement with more precise analytical models. However, the moment coefficient, given by Equa­tion 3.27, is only three-quarters of that obtained by analytical means. Figure 3.19 shows that the exact value is approached rapidly, however, as the number of segments increases.

As indicated by Figure 3.19, the exact value of the moment coefficient about the aerodynamic center (c/4) for the circular arc airfoil is given by

Стар 77" Z

Using Equation 3.13, the location of the center of pressure can be found as

N Figure 3.19 Numerical calculation of moment coefficient compared with analy­tical prediction for 4% cambered, circular arc airfoil.

Observe that as Q decreases, the center of pressure moves aft, ap­proaching infinity as C| goes to zero. This movement of the center of pressure is opposite to what was believed to be true by the early pioneers in aviation. The Wright Brothers were probably the first to recognize the true nature of the center-of-pressure movement as a result of their meticulous wind tunnel tests.

Analytical solutions to the thin airfoil can be found in several texts (e. g., Ref. 3.2 and 3.3). Here, the airfoil is replaced by a continuous distribution of vortices instead of discrete point vortices, as used with the numerical solu­tion.

Referring to Figure 3.20, without any loss of generality, the airfoil is taken to have a unit chord lying along the x-axis with the origin at the leading

% Figure 3.20 The modeling of a thin airfoil by a vortex sheet.

edge. The shape of the camber line is given by z(x), and it is assumed that

z(x)«1

With this assumption the problem is linearized and made tractable by replac­ing the airfoil with a vortex sheet of unit strength y(x) lying along the chord line instead of along the camber line.

At the point Xo, the downward velocity induced by an elemental vortex of strength y(x) dx located at x, according to Equation 2.56, will be given by

y(s) dx 2tt(Xq – x)

or, integrating over the chord,

In order to satisfy the boundary condition that the flow be tangent everywhere to the mean camber line, it follows that, to a small angle approximation,

(3.32)

Thus, given a and z(x), the following integral equation must be solved for y(x).

(3.33)

In addition, y(x) must vanish at the trailing edge in order to satisfy the Kutta condition. Otherwise, the induced velocity will be infinite just down­stream of this point.

Equation 3.33 is solved by first transforming to polar coordinates.

Letting

x=|(l-cos0) (3.34)

Equation 3.33 becomes

1 (w_JiWde_ = a_(dz 2ttV Jo cos в – cos во dx4)

On the basis of the more sophisticated method of conformal mapping (e. g., see Ref. 3.4), it is known that y(x) is generally singular at the leading edge approaching infinity as 1/x. Thus we will assume a priori that Equation 3.35 can be satisfied by а у(в) distribution of the form

Г „ (1 + COS0) . V’ „ • J

lA°–sine +2*sm»eJ

Using the relationships

| j^cos (n – 1)0 – cos (n + 1)0 J = sin n0 sin 0 and

f" cos nOde _ sin n0о Jo COS 0 – COS 00 sin 00

Equation 3.35 becomes

A0 – 2 A„ cos nO = at

n = l

Multiplying both sides of the preceding equation by cos m0 (m =0, 1,2,…, n,…) and integrating from 0 to тт leads to

A‘-a~H’£de <3-38a)

A„ = ^J ^cosn0d0 (3.38b)

Thus, knowing the shape of the mean camber line, the coefficients A0, Au A2,… can be determined either in closed form or by graphical or numerical means (see Ref. 3.1). Having these coefficients, Q and Cm can then be easily determined from the Kutta-Joukowski relationship.

The lift and moment about the leading edge are given by

L = f pVy(x) dx Jo

MLE= – CPVy(x)xdx Jo

From these and using Equation 3.36,

Q = 2-nAo + itAi (3.39)

C"LE=-f (ao + A,-^) (3.40)

It follows that Cm about the quarter-chord point is independent of a, so that this point is the aerodynamic center, with the moment coefficient being given by

Cmac = – f(A.-A2) (3.41)

Since a is contained only in the A0 coefficient, it can be concluded immediately without considering the actual form of z(x) that Q is given by a

linear combination of a and a function of z. Thus, camber changes can be expected to affect the angle of zero lift but not the slope of the lift curve.

Reference to airfoil data, such as that presented in Figures 3.5 and 3.6, will show that the predictions of thin airfoil theory are essentially correct. There is a range of angles of attack over which the lift coefficient varies linearly with a. The slope of this lift curve is usually not as high as the theory predicts, being approximately 4 to 8% less than the theoretical value. For many purposes an assumed value of 0.1 C(/deg is sufficiently accurate and is a useful number to remember. Experimental data also show the aerodynamic center to be close to the quarter-chord point. The effects of camber on Q, dQlda, and Cmac are also predicted well.

Recently large numerical programs have been developed to predict the performance of airfoils that incorporate Reynolds number and Mach number effects.’These are typified by Reference 3.5, which will be described briefly. This program begins by calculating the potential flow around the airfoil. In order to allow for both finite thickness and circulation, the airfoil contour is approximated by a closed polygon, as shown in Figure 3.21. A continuous distribution of vortices is then placed on each side of the polygon, with the vortex strength per unit length, y, varying linearly from one corner to the next and continuous across the corner. Figure 3.22 illustrates this model for two sides connecting corners 3, 4, and 5. Control points are chosen midway between the corners. The values of the vortex unit strengths at the corners are then found that will induce velocities at each control point tangent to the polygon side at that point. Note, however, that if there are n corners and hence n + 1 unknown у values at the corners, the n control points provide one less equation than unknowns. This situation is remedied by applying the Kutta condition at the trailing edge. This requires that yn+1 = – yb assuring that the velocities induced at the trailing edge are finite.

Flaving determined the vortex strengths, the velocity field and, hence, the

Figure 3.21 Approximation of airfoil contour by closed polygon.

Уз

4

Figure 3.22 Vortex distributions representing airfoil contour.

pressure distribution around the airfoil can be calculated. This result is then used to calculate the boundary layer development over the airfoil, including the growth of the laminar layer, transition, the growth of the turbulent layer, and possible boundary layer separation. The airfoil shape is then enlarged slightly to allow for the boundary layer thickness and the potential flow solutions are repeated. The details of this iterative procedure are beyond the scope of this text.